Due to the potentially strong adverse effect of surfactants on the drag reduction performance of SHSs, it is crucial to model their impact.
Theoretical models can explain experimental observations, and also provide predictions for the design of SHSs where surfactant effects can be mitigated.
However, as discussed by [1]}, the theoretical modelling of flows inclusive of surfactant over SHSs is complex.
The governing equations involve coupled nonlinear transport equations for the conservation of mass, momentum, bulk surfactant and interfacial surfactant in three-dimensional (3D) geometries with nonlinear boundary conditions at the interfaces.
Hence, analytical approaches can be challenging, whilst numerical simulations remain computationally intensive.
To make progress, [1]} assumed that the surfactant concentration is small.
This assumption is consistent with normal environmental conditions where surfactants are found in trace amounts [3]}.
[1]} constructed a scaling theory to model the slip and drag in steady two-dimensional (2D) pressure-driven channel flows bounded by SHSs, made of long transverse gratings, in the low-Reynolds-number flow regime.
They also performed finite-element 2D numerical simulations to compute the constants in the scaling theory, and validate their scaling predictions.
Their results showed that the slip length and drag reduction are affected by surfactants across a broad range of the parameter space.
However, their model was not rigorously derived from first principles, nor did they perform a full asymptotic analysis of the parameter space.
Moreover, their theory did not capture stagnant cap regimes, where the upstream region of the interface has a weak surfactant gradient and is almost shear-free, whilst the downstream region of the interface has a strong surfactant gradient and is effectively no-slip [5]}, [6]}, [7]}, .
In this regime, the advection of surfactant at the interface dominates relative to surfactant exchange between bulk and surface, leading to non-uniform interfacial concentration distributions.
